7 SEMICONTINUOUS FUNCTIONS 29 2. (a) Let E, F be two metrizable locally compact spaces and let TT: E->F be a con- tinuous map. The mapping TT is said to be proper if, for each compact subset K of F, the inverse image TT'^K) is compact. Show that, if TT is proper and y e 7r(E), every neighborhood in E of the set 7r~l(y) contains a neighborhood of the form TT'^U), where U is a neighborhood of y in F. (b) Let g be a lower semicontinuous real-valued function on E. For each y e F, let/0>) be the greatest lower bound of g on the set rr~l(y) (so that/(y) = +00 if rr~l(y) = 0). Show that /is lower semicontinuous on F. (c) Let g be the continuous function (xt, x2)t->\XiX2 1| on R x ]0, -f oo [. For each x e R, let/(*i) = inf g(xi9 x2). Show that/is not lower semicontinuous onR. *2>0 3. For each point / = (tl9 . . . , tn) e C", put P,(X) = X" + t^"1 + + /. Show that there exists a continuous mapping t\-^x(t) of C" into R and a lower semicontinuous mapping t \-+y(t) of C" into R, such that Pf(x(t) + iy(t)) = 0 for all t e C1. (Take x(t) to be the largest of the real parts of the roots of P, , and use (9.1 7.4)). 4. Let / be any mapping of a metric space E into a metric space F. For each x e E, let Łl(x) be the oscillation of /at x with respect to E (3.14), so that Łl(x) is a positive real number or + °° Show that the mapping x \ > Q(x) of E into R is upper semicontinuous. 5. Let E be a metric space and let/: E ->R be lower semicontinuous. Let <re E be a point at which the oscillation Q,(a) of / is finite. Show that for each e > 0, there exists a neighborhood V of a in E such that inf Łl(x) fŁ e. (Show that otherwise there would X 6 V exist points x arbitrarily close to a at which /(#) took arbitrarily large values.) 6. Let (fl,,)oi be an infinite sequence of distinct points in the interval [0, 1[. For each integer N>1, let bi, ..., b^ denote the sequence obtained by arranging the set {fli, ...,0N} in increasing order. The intervals [0, ŁJ, [bi9b2[9...9 [6N-i, ŁN[»[^N, H are said to form the NT//z subdivision of [0, 1 [ corresponding to the sequence (an). Let WN (resp. v^) denote the minimum (resp. maximum) of the lengths of the intervals of the Nth subdivision. Let A = lim inf NJ/N , JJL = lim sup NuN . N-+OO N-+OO (a) For each e > 0, let n0 be such that #N ;> (A - e)/N and yN <; (/* + e)/N for all N ^ /?0 Show that, for each N ^> «0 and each integer r such that 0 ^ r ^ N» there exist 2r intervals of the (N + r)th subdivision whose lengths are greater than or equal to the 2r numbers X s A e A e A e A e A e NTT' N + 2' N + 2''N-f-rf N+7 and such that the other intervals of the (N + r)th subdivision are intervals of the Nth subdivision (use induction on r). (b) Show that, if N §: nQ , the intervals of the Nth subdivision can be arranged in order in such a way that their lengths are less than or equal to the N + 1 numbers p+'e N ' N.+ 1''"' 2N ' (For each interval I of the Nth subdivision, let s(l) be the largest integer ^2N such that I is an interval of the s(I)th subdivision, and arrange the intervals I in order of increasing .s(I).)(cf. (3.13.3) and (3.13.4)). To get corresponding results in general, it is